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The Sample Proportion The letter p represents the population proportion. The symbol p ^ (“p-hat”) represents the sample proportion. p ^ is a random variable. The sampling distribution of p ^ is the probability distribution of all the possible values of p ^.

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Example Suppose that 2/3 of all males wash their hands after using a public restroom. Suppose that we take a sample of 1 male. Find the sampling distribution of p ^.

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Example W N 2/3 1/3 P(W) = 2/3 P(N) = 1/3

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Example Let x be the sample number of males who wash. The probability distribution of x is xP(x)P(x) 01/3 12/3

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Our Experiment In our experiment, we had 80 samples of size 5. Based on the sampling distribution when n = 5, we would expect the following Value of p ^ Actual Predicted

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The pdf when n = 1 01

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The pdf when n = 2 011/2

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The pdf when n = 3 011/32/3

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The pdf when n = 4 011/42/43/4

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The pdf when n = 5 011/52/53/5 4/5

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1 8/10 The pdf when n = 10 02/104/106/10

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Observations and Conclusions Observation: The values of p ^ are clustered around p. Conclusion: p ^ is close to p most of the time.

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Observations and Conclusions Observation: As the sample size increases, the clustering becomes tighter. Conclusion: Larger samples give better estimates. Conclusion: We can make the estimates of p as good as we want, provided we make the sample size large enough.

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Observations and Conclusions Observation: The distribution of p ^ appears to be approximately normal. Conclusion: We can use the normal distribution to calculate just how close to p we can expect p ^ to be.

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One More Observation However, we must know the values of  and  for the distribution of p ^. That is, we have to quantify the sampling distribution of p ^.

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The Central Limit Theorem for Proportions It turns out that the sampling distribution of p ^ is approximately normal with the following parameters.

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The Central Limit Theorem for Proportions The approximation to the normal distribution is excellent if

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Example If we gather a sample of 100 males, how likely is it that between 60 and 70 of them, inclusive, wash their hands after using a public restroom? This is the same as asking the likelihood that 0.60  p ^  0.70.

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Why Surveys Work Suppose that we are trying to estimate the proportion of the male population who wash their hands after using a public restroom. Suppose the true proportion is 66%. If we survey a random sample of 1000 people, how likely is it that our error will be no greater than 5%?

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Why Surveys Work Now we have

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Why Surveys Work Now find the probability that p^ is between 0.61 and 0.71: normalcdf(.61,.71,.66,.01498) = It is virtually certain that our estimate will be within 5% of 66%.

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Why Surveys Work What if we had decided to save money and surveyed only 100 people? If it is important to be within 5% of the correct value, is it worth it to survey 1000 people instead of only 100 people?

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Quality Control A company will accept a shipment of components if there is no strong evidence that more than 5% of them are defective. H 0 : 5% of the parts are defective. H 1 : More than 5% of the parts are defective.

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Quality Control They will take a random sample of 100 parts and test them. If no more than 10 of them are defective, they will accept the shipment. What is  ? What is  ?